The Syrian army told the insurgents to leave their positions, offering safe passage and aid supplies.
Syrian forces supported by Iranian-backed militias and Russian air power began their push to take the whole of the divided city after a ceasefire collapsed last month.
An air campaign by the Syrian government and its allies has been reinforced by a ground offensive against the besieged eastern half of Aleppo, where insurgents have been holding out. Hospitals have been badly hit in the assault, medics say.
While Russian Foreign Minister Sergei Lavrov and US Secretary of State John Kerry spoke by phone to discuss normalization of the situation, Britain said the bombing of hospitals by forces loyal to Syrian President Bashar al Assad made it impossible talk about peace.
"It is the continuing savagery of the Assad regime against the people of Aleppo and the complicity of the Russians in committing what are patently war crimes - bombing hospitals, when they know they are hospitals and nothing but hospitals - that is making it impossible for peace negotiations to resume," British Foreign Secretary Boris Johnson said.
The Syrian Observatory for Human Rights and the Syrian military said on Sunday that the army and its allies had advanced south from the Handarat refugee camp north of the city, taking the Kindi hospital and parts of the Shuqaif industrial area.